Many large finite groups \(G\)  exhibit expansion and mixing phenomena. The former states that product sets \(A_1A_2\subseteq G\)  are always significantly larger than either of \(A_1,A_2\subseteq G\) . The latter states that when \(A_1,A_2,A_3\subseteq G\)  are fairly large, then the number of solutions to \(a_1a_2=a_3\)  for \(a_i\in A_i\)  is close to \(\frac{|A_1A_2A_3|}{|G|^2}.\)  Such properties have attracted a lot of interest and are related to theoretical computer science via the notion of expander graphs [1]}, [2]}. Seminal papers have established such properties for certain simple groups, see for instance [3]}, [4]}, [5]}, [6]}, [7]}.
